Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(9,10,13,16)(11,14,15,12)(19,23)(21,24);(9,10)(11,14)(12,15)(13,16)(17,18,20,22)(19,21,23,24);(5,6,8,7)(9,11)(10,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24);(1,2,3,4)(5,7,8,6)(9,11)(10,14)(12,16)(13,15);(9,12,13,14)(10,15,16,11)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fg,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4342422);